Theorem qsqeqor 10630

Description: The squares of two rational numbers are equal iff one number equals the other or its negative. (Contributed by Jim Kingdon, 1-Nov-2024.)

Assertion

Ref	Expression
qsqeqor	⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)))

Proof of Theorem qsqeqor

Step	Hyp	Ref	Expression
1		qre 9624	. . . . . . 7 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℝ)
2	1	ad3antrrr 492	. . . . . 6 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 0 ≤ 𝐵) → 𝐴 ∈ ℝ)
3		simplr 528	. . . . . 6 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 0 ≤ 𝐵) → 0 ≤ 𝐴)
4		qre 9624	. . . . . . 7 ⊢ (𝐵 ∈ ℚ → 𝐵 ∈ ℝ)
5	4	ad3antlr 493	. . . . . 6 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 0 ≤ 𝐵) → 𝐵 ∈ ℝ)
6		simpr 110	. . . . . 6 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 0 ≤ 𝐵) → 0 ≤ 𝐵)
7		sq11 10592	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → ((𝐴↑2) = (𝐵↑2) ↔ 𝐴 = 𝐵))
8	2, 3, 5, 6, 7	syl22anc 1239	. . . . 5 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 0 ≤ 𝐵) → ((𝐴↑2) = (𝐵↑2) ↔ 𝐴 = 𝐵))
9		orc 712	. . . . 5 ⊢ (𝐴 = 𝐵 → (𝐴 = 𝐵 ∨ 𝐴 = -𝐵))
10	8, 9	syl6bi 163	. . . 4 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 0 ≤ 𝐵) → ((𝐴↑2) = (𝐵↑2) → (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)))
11		oveq1 5881	. . . . . . 7 ⊢ (𝐴 = 𝐵 → (𝐴↑2) = (𝐵↑2))
12	11	a1i 9	. . . . . 6 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 = 𝐵 → (𝐴↑2) = (𝐵↑2)))
13		oveq1 5881	. . . . . . . . 9 ⊢ (𝐴 = -𝐵 → (𝐴↑2) = (-𝐵↑2))
14	13	adantl 277	. . . . . . . 8 ⊢ (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 = -𝐵) → (𝐴↑2) = (-𝐵↑2))
15		qcn 9633	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℚ → 𝐵 ∈ ℂ)
16		sqneg 10578	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℂ → (-𝐵↑2) = (𝐵↑2))
17	15, 16	syl 14	. . . . . . . . 9 ⊢ (𝐵 ∈ ℚ → (-𝐵↑2) = (𝐵↑2))
18	17	ad2antlr 489	. . . . . . . 8 ⊢ (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 = -𝐵) → (-𝐵↑2) = (𝐵↑2))
19	14, 18	eqtrd 2210	. . . . . . 7 ⊢ (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 = -𝐵) → (𝐴↑2) = (𝐵↑2))
20	19	ex 115	. . . . . 6 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (𝐴 = -𝐵 → (𝐴↑2) = (𝐵↑2)))
21	12, 20	jaod 717	. . . . 5 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → ((𝐴 = 𝐵 ∨ 𝐴 = -𝐵) → (𝐴↑2) = (𝐵↑2)))
22	21	ad2antrr 488	. . . 4 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 0 ≤ 𝐵) → ((𝐴 = 𝐵 ∨ 𝐴 = -𝐵) → (𝐴↑2) = (𝐵↑2)))
23	10, 22	impbid 129	. . 3 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 0 ≤ 𝐵) → ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)))
24	17	eqeq2d 2189	. . . . . 6 ⊢ (𝐵 ∈ ℚ → ((𝐴↑2) = (-𝐵↑2) ↔ (𝐴↑2) = (𝐵↑2)))
25	24	ad3antlr 493	. . . . 5 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → ((𝐴↑2) = (-𝐵↑2) ↔ (𝐴↑2) = (𝐵↑2)))
26	1	ad3antrrr 492	. . . . . . 7 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → 𝐴 ∈ ℝ)
27		simplr 528	. . . . . . 7 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → 0 ≤ 𝐴)
28		qnegcl 9635	. . . . . . . . 9 ⊢ (𝐵 ∈ ℚ → -𝐵 ∈ ℚ)
29		qre 9624	. . . . . . . . 9 ⊢ (-𝐵 ∈ ℚ → -𝐵 ∈ ℝ)
30	28, 29	syl 14	. . . . . . . 8 ⊢ (𝐵 ∈ ℚ → -𝐵 ∈ ℝ)
31	30	ad3antlr 493	. . . . . . 7 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → -𝐵 ∈ ℝ)
32		simpr 110	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → 𝐵 ≤ 0)
33	4	le0neg1d 8473	. . . . . . . . 9 ⊢ (𝐵 ∈ ℚ → (𝐵 ≤ 0 ↔ 0 ≤ -𝐵))
34	33	ad3antlr 493	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → (𝐵 ≤ 0 ↔ 0 ≤ -𝐵))
35	32, 34	mpbid 147	. . . . . . 7 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → 0 ≤ -𝐵)
36		sq11 10592	. . . . . . 7 ⊢ (((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) ∧ (-𝐵 ∈ ℝ ∧ 0 ≤ -𝐵)) → ((𝐴↑2) = (-𝐵↑2) ↔ 𝐴 = -𝐵))
37	26, 27, 31, 35, 36	syl22anc 1239	. . . . . 6 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → ((𝐴↑2) = (-𝐵↑2) ↔ 𝐴 = -𝐵))
38		olc 711	. . . . . 6 ⊢ (𝐴 = -𝐵 → (𝐴 = 𝐵 ∨ 𝐴 = -𝐵))
39	37, 38	syl6bi 163	. . . . 5 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → ((𝐴↑2) = (-𝐵↑2) → (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)))
40	25, 39	sylbird 170	. . . 4 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → ((𝐴↑2) = (𝐵↑2) → (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)))
41	21	ad2antrr 488	. . . 4 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → ((𝐴 = 𝐵 ∨ 𝐴 = -𝐵) → (𝐴↑2) = (𝐵↑2)))
42	40, 41	impbid 129	. . 3 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) ∧ 𝐵 ≤ 0) → ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)))
43		0z 9263	. . . . . 6 ⊢ 0 ∈ ℤ
44		zq 9625	. . . . . 6 ⊢ (0 ∈ ℤ → 0 ∈ ℚ)
45	43, 44	ax-mp 5	. . . . 5 ⊢ 0 ∈ ℚ
46		qletric 10243	. . . . 5 ⊢ ((0 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (0 ≤ 𝐵 ∨ 𝐵 ≤ 0))
47	45, 46	mpan 424	. . . 4 ⊢ (𝐵 ∈ ℚ → (0 ≤ 𝐵 ∨ 𝐵 ≤ 0))
48	47	ad2antlr 489	. . 3 ⊢ (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) → (0 ≤ 𝐵 ∨ 𝐵 ≤ 0))
49	23, 42, 48	mpjaodan 798	. 2 ⊢ (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 0 ≤ 𝐴) → ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)))
50		qnegcl 9635	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℚ → -𝐴 ∈ ℚ)
51		qre 9624	. . . . . . . . . 10 ⊢ (-𝐴 ∈ ℚ → -𝐴 ∈ ℝ)
52	50, 51	syl 14	. . . . . . . . 9 ⊢ (𝐴 ∈ ℚ → -𝐴 ∈ ℝ)
53	52	ad3antrrr 492	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → -𝐴 ∈ ℝ)
54		simplr 528	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → 𝐴 ≤ 0)
55	1	le0neg1d 8473	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℚ → (𝐴 ≤ 0 ↔ 0 ≤ -𝐴))
56	55	ad3antrrr 492	. . . . . . . . 9 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → (𝐴 ≤ 0 ↔ 0 ≤ -𝐴))
57	54, 56	mpbid 147	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → 0 ≤ -𝐴)
58	4	ad3antlr 493	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → 𝐵 ∈ ℝ)
59		simpr 110	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → 0 ≤ 𝐵)
60		sq11 10592	. . . . . . . 8 ⊢ (((-𝐴 ∈ ℝ ∧ 0 ≤ -𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) → ((-𝐴↑2) = (𝐵↑2) ↔ -𝐴 = 𝐵))
61	53, 57, 58, 59, 60	syl22anc 1239	. . . . . . 7 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → ((-𝐴↑2) = (𝐵↑2) ↔ -𝐴 = 𝐵))
62	61	biimpd 144	. . . . . 6 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → ((-𝐴↑2) = (𝐵↑2) → -𝐴 = 𝐵))
63		qcn 9633	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℚ → 𝐴 ∈ ℂ)
64		sqneg 10578	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℂ → (-𝐴↑2) = (𝐴↑2))
65	63, 64	syl 14	. . . . . . . . 9 ⊢ (𝐴 ∈ ℚ → (-𝐴↑2) = (𝐴↑2))
66	65	adantr 276	. . . . . . . 8 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (-𝐴↑2) = (𝐴↑2))
67	66	eqeq1d 2186	. . . . . . 7 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → ((-𝐴↑2) = (𝐵↑2) ↔ (𝐴↑2) = (𝐵↑2)))
68	67	ad2antrr 488	. . . . . 6 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → ((-𝐴↑2) = (𝐵↑2) ↔ (𝐴↑2) = (𝐵↑2)))
69		negcon1 8208	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (-𝐴 = 𝐵 ↔ -𝐵 = 𝐴))
70	63, 15, 69	syl2an 289	. . . . . . . 8 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (-𝐴 = 𝐵 ↔ -𝐵 = 𝐴))
71		eqcom 2179	. . . . . . . 8 ⊢ (-𝐵 = 𝐴 ↔ 𝐴 = -𝐵)
72	70, 71	bitrdi 196	. . . . . . 7 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (-𝐴 = 𝐵 ↔ 𝐴 = -𝐵))
73	72	ad2antrr 488	. . . . . 6 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → (-𝐴 = 𝐵 ↔ 𝐴 = -𝐵))
74	62, 68, 73	3imtr3d 202	. . . . 5 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → ((𝐴↑2) = (𝐵↑2) → 𝐴 = -𝐵))
75	74, 38	syl6 33	. . . 4 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → ((𝐴↑2) = (𝐵↑2) → (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)))
76	21	ad2antrr 488	. . . 4 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → ((𝐴 = 𝐵 ∨ 𝐴 = -𝐵) → (𝐴↑2) = (𝐵↑2)))
77	75, 76	impbid 129	. . 3 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 0 ≤ 𝐵) → ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)))
78	52	ad3antrrr 492	. . . . . . 7 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → -𝐴 ∈ ℝ)
79		simplr 528	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → 𝐴 ≤ 0)
80	55	ad3antrrr 492	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → (𝐴 ≤ 0 ↔ 0 ≤ -𝐴))
81	79, 80	mpbid 147	. . . . . . 7 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → 0 ≤ -𝐴)
82	30	ad3antlr 493	. . . . . . 7 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → -𝐵 ∈ ℝ)
83		simpr 110	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → 𝐵 ≤ 0)
84	33	ad3antlr 493	. . . . . . . 8 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → (𝐵 ≤ 0 ↔ 0 ≤ -𝐵))
85	83, 84	mpbid 147	. . . . . . 7 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → 0 ≤ -𝐵)
86		sq11 10592	. . . . . . 7 ⊢ (((-𝐴 ∈ ℝ ∧ 0 ≤ -𝐴) ∧ (-𝐵 ∈ ℝ ∧ 0 ≤ -𝐵)) → ((-𝐴↑2) = (-𝐵↑2) ↔ -𝐴 = -𝐵))
87	78, 81, 82, 85, 86	syl22anc 1239	. . . . . 6 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → ((-𝐴↑2) = (-𝐵↑2) ↔ -𝐴 = -𝐵))
88	65, 17	eqeqan12d 2193	. . . . . . 7 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → ((-𝐴↑2) = (-𝐵↑2) ↔ (𝐴↑2) = (𝐵↑2)))
89	88	ad2antrr 488	. . . . . 6 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → ((-𝐴↑2) = (-𝐵↑2) ↔ (𝐴↑2) = (𝐵↑2)))
90	63	ad3antrrr 492	. . . . . . 7 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → 𝐴 ∈ ℂ)
91	15	ad3antlr 493	. . . . . . 7 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → 𝐵 ∈ ℂ)
92	90, 91	neg11ad 8263	. . . . . 6 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → (-𝐴 = -𝐵 ↔ 𝐴 = 𝐵))
93	87, 89, 92	3bitr3d 218	. . . . 5 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → ((𝐴↑2) = (𝐵↑2) ↔ 𝐴 = 𝐵))
94	93, 9	syl6bi 163	. . . 4 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → ((𝐴↑2) = (𝐵↑2) → (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)))
95	21	ad2antrr 488	. . . 4 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → ((𝐴 = 𝐵 ∨ 𝐴 = -𝐵) → (𝐴↑2) = (𝐵↑2)))
96	94, 95	impbid 129	. . 3 ⊢ ((((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) ∧ 𝐵 ≤ 0) → ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)))
97	47	ad2antlr 489	. . 3 ⊢ (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) → (0 ≤ 𝐵 ∨ 𝐵 ≤ 0))
98	77, 96, 97	mpjaodan 798	. 2 ⊢ (((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) ∧ 𝐴 ≤ 0) → ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)))
99		qletric 10243	. . . 4 ⊢ ((0 ∈ ℚ ∧ 𝐴 ∈ ℚ) → (0 ≤ 𝐴 ∨ 𝐴 ≤ 0))
100	45, 99	mpan 424	. . 3 ⊢ (𝐴 ∈ ℚ → (0 ≤ 𝐴 ∨ 𝐴 ≤ 0))
101	100	adantr 276	. 2 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → (0 ≤ 𝐴 ∨ 𝐴 ≤ 0))
102	49, 98, 101	mpjaodan 798	1 ⊢ ((𝐴 ∈ ℚ ∧ 𝐵 ∈ ℚ) → ((𝐴↑2) = (𝐵↑2) ↔ (𝐴 = 𝐵 ∨ 𝐴 = -𝐵)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 708   = wceq 1353   ∈ wcel 2148   class class class wbr 4003  (class class class)co 5874  ℂcc 7808  ℝcr 7809  0cc0 7810   ≤ cle 7992  -cneg 8128  2c2 8969  ℤcz 9252  ℚcq 9618  ↑cexp 10518

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-mulrcl 7909  ax-addcom 7910  ax-mulcom 7911  ax-addass 7912  ax-mulass 7913  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-1rid 7917  ax-0id 7918  ax-rnegex 7919  ax-precex 7920  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-apti 7925  ax-pre-ltadd 7926  ax-pre-mulgt0 7927  ax-pre-mulext 7928

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-ilim 4369  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-frec 6391  df-pnf 7993  df-mnf 7994  df-xr 7995  df-ltxr 7996  df-le 7997  df-sub 8129  df-neg 8130  df-reap 8531  df-ap 8538  df-div 8629  df-inn 8919  df-2 8977  df-n0 9176  df-z 9253  df-uz 9528  df-q 9619  df-rp 9653  df-seqfrec 10445  df-exp 10519

This theorem is referenced by:  4sqlem10  12384